Difficult Problems and Why We Like Them

Audrey Weeks writes a really great set of animations for calculus teachers and students called Calculus in Motion. The animations run on Geometer’s Sketchpad and are very easy to use, but difficult to program. But Audrey is an expert. She is very busy each year at this time doing animations of the recent AP calculus exam questions. I proofread the animations as she finishes each one and now and then make some suggestions.

The particular solution to the differential equation question BC 5 on this year’s exam is a function with a vertical asymptote. The animation allows the user to move the initial condition around. This moves the asymptote(s); in some cases there is no asymptote.Her graph showed the function on both sides of the asymptote with the note that the domain was x > –1.I pointed out that the graph should only exist on one side of the asymptote at x = –1. She wrote back that “I agree, and I had tried, but couldn’t find any way to do that.”

Then the next day she sent the next version with only the correct part of the graph showing, and its changing restricted domain stated as the initial condition changed.

See the figure below with no graph to the left of the asymptote x = –1

She wrote:

“Now, will anyone appreciate the additional 5 hours of work to make that happen, …? Doesn’t matter … it makes ME happy.”

Item 2:

Kryptos is a sculpture by artist Jim Sanborn that stands on the campus of the Central Intelligence Agency (CIA) Headquarters in Langley, VA. The sculpture contains four coded messages.

In a recent article Wired.com recounts how David Stein, a CIA employee who is not a cryptographer, got interested in the cipher and spent 400 hours over 8 years trying to break it. He succeeded in deciphering three of the four messages in 1998. He did the deciphering with pencil-and-paper in his spare time.

The CIA would not allow him to make his results public at the time. (Later, Jim Gillogly, a computer scientist, cracked the same 3 messages using a computer and, not being employed by the CIA, published his results.) The CIA recently declassified Stein work.

The Wired article recounts the story when (quoting Stein), “I was I was hit by that sweetly ecstatic, rare experience that I have heard described as a ‘moment of clarity.’ All the doubts and speculations about the thousands of possible alternate paths simply melted away, and I clearly saw the one correct course laid out in front of me.” Stein’s full article is included in the Wired article (photocopied, slightly redacted, and missing the figures and charts). The solution is here. To this day no one has deciphered the fourth message.

Stein’s account concludes with this remark:

When confronted with a puzzle or problem, we sometimes can lose sight of the fact that we have issued a challenge to ourselves–not to our tools. And before we automatically reach for our computers, we sometimes need to remember that we already possess the most essential and powerful problem-solving tool within our own minds.

Update (November 21, 2014) The sculptor of Kryptos has provide a second clue to the fourth panel of the sculptor. The full story is here. The full enciphered text is below. The source (N.Y. Times November 21, 2014) with the known deciphering and the clues is here.